def zeta_series(self, n, t):
"""
Return the zeta series.
Compute a power series approximation to the zeta function of a
scheme over a finite field.
INPUT:
- ``n`` -- the number of terms of the power series to
compute
- ``t`` -- the variable which the series should be
returned
OUTPUT:
A power series approximating the zeta function of self
EXAMPLES::
sage: P.<x> = PolynomialRing(GF(3))
sage: C = HyperellipticCurve(x^3+x^2+1)
sage: R.<t> = PowerSeriesRing(Integers())
sage: C.zeta_series(4,t)
1 + 6*t + 24*t^2 + 78*t^3 + 240*t^4 + O(t^5)
sage: (1+2*t+3*t^2)/(1-t)/(1-3*t) + O(t^5)
1 + 6*t + 24*t^2 + 78*t^3 + 240*t^4 + O(t^5)
Note that this function depends on count_points, which is only
defined for prime order fields for general schemes.
Nonetheless, since :trac:`15108` and :trac:`15148`, it supports
hyperelliptic curves over non-prime fields::
sage: C.base_extend(GF(9,'a')).zeta_series(4,t)
1 + 12*t + 120*t^2 + 1092*t^3 + 9840*t^4 + O(t^5)
"""
F = self.base_ring()
if not F.is_finite():
raise TypeError(
'zeta functions only defined for schemes over finite fields')
try:
a = self.count_points(n)
except AttributeError:
raise NotImplementedError(
'count_points() required but not implemented')
R = PowerSeriesRing(Rationals(), 'u')
u = R.gen()
temp = sum(a[i - 1] * (u.O(n + 1))**i / i for i in range(1, n + 1))
temp2 = temp.exp()
return (temp2(t).O(n + 1))
def zeta_series(self, n, t):
"""
Return the zeta series.
Compute a power series approximation to the zeta function of a
scheme over a finite field.
INPUT:
- ``n`` - the number of terms of the power series to
compute
- ``t`` - the variable which the series should be
returned
OUTPUT:
A power series approximating the zeta function of self
EXAMPLES::
sage: P.<x> = PolynomialRing(GF(3))
sage: C = HyperellipticCurve(x^3+x^2+1)
sage: R.<t> = PowerSeriesRing(Integers())
sage: C.zeta_series(4,t)
1 + 6*t + 24*t^2 + 78*t^3 + 240*t^4 + O(t^5)
sage: (1+2*t+3*t^2)/(1-t)/(1-3*t) + O(t^5)
1 + 6*t + 24*t^2 + 78*t^3 + 240*t^4 + O(t^5)
Note that this function depends on count_points, which is only
defined for prime order fields::
sage: C.base_extend(GF(9,'a')).zeta_series(4,t)
Traceback (most recent call last):
...
NotImplementedError: Point counting only implemented for schemes over prime fields
"""
F = self.base_ring()
if not F.is_finite():
raise TypeError, "Zeta functions only defined for schemes over finite fields"
a = self.count_points(n)
R = PowerSeriesRing(Rationals(), 'u')
u = R.gen()
temp = sum(a[i - 1] * (u.O(n + 1))**i / i for i in range(1, n + 1))
temp2 = temp.exp()
return (temp2(t).O(n + 1))